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Evaluate: lim (x→0) ((cos 9x – cos 7x)/(cos 3x – cos5x)) |
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Answer» \(\lim\limits_{x \to 10} \) \(\cfrac{cos9x-cos7x}{cos3x-cos5x}\) (\(\frac{0}{0}\)type) = \(\lim\limits_{x \to 10} \) \(\cfrac{-9sin9x +7sin7x}{-3sin3x+5sinx5x}\) (By applying D.L.H Rule) = \(\lim\limits_{x \to 10} \) \(\cfrac{-9\frac{sin9x}{9x}\times9x+7\frac{sin7x}{7x}\times7x}{-3\frac{sin3x}{3x}\times3x+5\frac{sin5x}{5x}\times5x}\) = \(\lim\limits_{x \to 10} \) \(\cfrac{x\Big(-81\frac{sin9x}{9x}+49\frac{sin7x}{7x}\Big)}{x\Big(-9\frac{sin3x}{3x}+25\frac{sin5x}{5x}\Big)}\) = \(\cfrac{-81\lim\limits_{x \to 10}\frac{sin9x}{9x}+49\lim\limits_{x \to10}\frac{sin7x}{7x}}{-9\lim\limits_{x \to 10}\frac{sin3x}{3x}+25\lim\limits_{x \to 10}\frac{sin5x}{5x}}\) = \(\frac{-81+49}{-91+25}\) = \(\frac{-32}{16}\) = -2 (∵\(\lim\limits_{x \to 10}\) \(\frac{sinax}{ax}\) = 1) |
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